An A320 Pilot completed an autoland but failed to turn the autopilot off after landing. While exiting the runway the aircraft attempted to turn back toward the runway with a jerking motion.

Narrative

After completing an autoland and clearing the runway on the P exit; the aircraft swerved right; then left and almost went off the left side of P. Although I did not believe we had left the pavement or hit anything (specifically taxi edge lights); we notified Ground to have the taxiway inspected to make sure. The aircraft was easily turned back to centerline and I felt it was safe to taxi to the gate. I called Dispatch after landing and let him know what happened and got the phone number of Tower. The Tower Supervisor informed me that the Operation's vehicle observed skid marks but there was no damage and nothing had left the pavement. Maintenance saw no damage to the aircraft or any evidence that we left the pavement. We did the autoland here because the autoland on the previous leg was marginal. The plane touched down right of centerline and 'hunted' to get back (the inflight portion of the approach was fine). I disconnected the autopilot and we taxied to the gate without a problem. My First Officer and I discussed whether to rate the autoland as SAT or UNSAT and decided that since we were taking the plane back to a maintenance station; we would attempt another autoland and see how the plane performed. At our destination; although we touched down right of centerline; it was very close and the rollout was smooth. In both cases; I had the brakes set to low and there was a slight right crosswind. I left the autopilot on a little longer than I have done in the past; but after coming out of reverse thrust and applying brake pressure to release the autobrakes; I thought I had disconnected the autopilot; using the control stick disconnect button (Later; I realized I didn't hear the disconnect caution chime). The turn to exit the runway at P seemed normal and I didn't seem to be fighting the airplane. After clearing the Runway; I configured the exterior lights for taxi and about this time; the plane swerved right; then left. The left turn might have been my reaction on the brakes; rudder and tiller to stop the swerve. During this time; my First Officer said something like; 'Hey; the autopilot is still on.' I think we both tried to disconnect because I heard 'priority left.' In any case; the autopilot disconnected and we stopped the plane. Although I don't think we were going very fast; both my First Officer and I were hard on the brakes to stop. In retrospect; while the Airbus will do some weird things; I don't think the autopilots came back on; so I didn't get a disconnect even though I thought I did. What I do not understand is how I was able; so easily; to turn off the Runway centerline to the taxiway without the autopilot fighting to stay on course. When I first checked out on the plane; I remember trying to steer before turning off the autopilot; it was not easy and very evident what was wrong. There was none of that feedback this time until we were on the taxiway. In the future; I will add the disconnect time to my approach briefing to both bring the First Officer more fully into the rollout loop (no pun intended) and also raise my own awareness of the last part of this procedure. I will also brief to listen for the disconnect chime.